• In the following slides I will try to evaluate what kind relationship that best fits the association between weight and miles pr. gallon in the mtcars dataset.

• I will fit three univariate linear regression models with the miles pr. gallon as the response and the weight, weight squared and log weight as the predictors

• I will then try to evaluate which one gives the best prediction based on a grphical evaluation and based on the

##A function to calculate the mean squared error
mse <- function(fit) mean((summary(fit)$residuals)^2)

##Lodaing packages and modifying the mtcars dataset
library(plotly); library(dplyr); library(tidyr)
mtcars_mod <- mtcars %>%
    mutate(log_wt = log(wt),
           squared_wt = wt^2)

#Fitting the models as described above
fit_lin <- lm(mpg ~ wt, data = mtcars_mod)
fit_sq <- lm(mpg ~ squared_wt, data = mtcars_mod)
fit_log <- lm(mpg ~ log_wt, data = mtcars_mod)

f <- list(
  family = "Courier New, monospace",
  size = 18,
  color = "#7f7f7f"
)
x <- list(
  title = "Weight (ton)",
  titlefont = f
)
y <- list(
  title = "Miles pr gallon (km)",
  titlefont = f
)

plot_ly(data = mtcars_mod, x = ~wt, y = ~mpg, 
        type = "scatter", 
        mode = "markers", 
        name = "Observed") %>%
    add_lines(x = ~wt, y = predict(fit_lin), 
              name = "Linear prediction") %>%
    add_lines(x = ~wt, y = predict(fit_sq), 
              name = "Squared prediction") %>%
    add_lines(x = ~wt, y = predict(fit_log), 
              name = "Logarithmic prediction") %>%
    layout(xaxis = x,
           yaxis = y)

From the graph it looks like the best prediction of miles pr. gallon comes from taking the log of weight and making a fit based on that. This is also indicated by the mean squared error which is, 8.7 for the linear prediction, 12.59 for the squared prediction and 6.68 for the logarithmic prediction.